I have an expression $\frac{d f}{d q}$ that I need in terms of $\frac{df}{dq_\star}$, so I need the expression $\frac{d q}{d q_\star}$. Great. In my situation $q=q(m_1, m_2)=m_2/m_1$ and $q_\star=q_\star(m_{1\star}, m_{2\star})=m_{2\star}/m_{1\star}$, and finally $m_1=m_1(m_{1\star})$ and $m_2=m_2(m_{2\star})$.
Using the derivative and chain rule,
$$ \frac{d q}{d q_\star} = \frac{\partial q}{\partial m_1}\frac{\partial m_1}{\partial m_{1\star}}\frac{\partial m_{1\star}}{\partial q_\star} + \frac{\partial q}{\partial m_2}\frac{\partial m_2}{\partial m_{2\star}}\frac{\partial m_{2\star}}{\partial q_\star}. $$
Then if I utilize the particular form of $q$ and $q_\star$ (quotients of the $m$ and $m_\star$ terms), then I (think I) can write this as:
$$ \frac{d q}{d q_\star} = \frac{m_{1\star}}{m_1} \left[ \frac{\partial m_2}{\partial m_{2\star}} + \frac{q}{q_\star} \frac{\partial m_1}{\partial m_{1\star}}\right]. $$
That seems reasonable to me.
But the problem is, if I consider a particular example in which $m_1 = a \, {m_{1\star}}^b$ and $m_2 = a \, {m_{1\star}}^b$ for two constants $a$ and $b$. If I plug-in to the expression above, then each term on the right-hand side gives me a $b \frac{q}{q_\star}$, so I finally end up with $$ \frac{d q}{d q_\star} \stackrel{?}{=} 2 b \frac{q}{q_\star}. $$ BUT, for this case, I can obviously write $q = {q_\star}^b$, which then obviously gives, $$ \frac{d q}{d q_\star} \stackrel{?}{=} b \frac{q}{q_\star}. $$
Why do these disagree by the factor of two? This seems like an incredibly simple problem, so I must be doing something trivially wrong here!
This definitely reminds me of the old "canceling-out differentials" issue where one might try: $$ df/dr = (\partial f / \partial x)(\partial x / \partial r) + (\partial f / \partial y)(\partial y / \partial r) = (\partial f / \partial r)(\partial x / \partial x) + (\partial f / \partial r)(\partial y / \partial y) = 2 (\partial f / \partial r). $$ Does that imply that I'm mixing up partial and complete derivatives somewhere?
This question is almost identical, but the answer given is basically that the meaning of "$dq/dq_\star$" is ambiguous. However, in this case $f$, $q$ and $q_\star$ have clear physical meanings and it (physically) seems very reasonable to ask how to translate from "how $f$ varies with $q$" to "how $f$ varies with $q_\star$", using "how $q$ varies with $q_\star$".